I would continue looking for the specific article that covers the procedure for the amendment of the bylaws, and I would not assume that the general rule about motions applies here. I have seen bylaws where the board was in charge of absolutely everything, with the exception of voting for board members, and moving to amend the bylaws.

Yes.
It is quite typical that the bylaws amendment process is special in one or more ways, compared to an ordinary main motion. I'd say it's more likely than not that they would be special in some way.
Also, the OP does not tell us that the bylaws provide that only the board may make motions. It was said to be their "practice", which is quite another thing. My guess would be that the board, over a period of time, has asserted certain powers that it did not actually have per the bylaws, and it became something of a custom to allow them to run the annual meeting. Goodness knows we've encountered this sort of thing here more times than can be counted on a shop teacher's fingers. My guess might be wrong, but for now it remains my guess.
The OP says that there is nothing in the bylaws that allows members to make motions at the AGM. And why would there be? Members have this right as a fundamental right of membership. Has anyone seen bylaws that "allows" members to make motions? The question is, do the bylaws prohibit members from making motions at the AGM? If not, they have this right already and nothing need be added to the bylaws. All that is required is that members know the rights they already have.
And if the bylaws actually do prohibit members from making motions, the appropriate amendment, it seems to me, would not be to add language allowing motions, but rather to strike the language that prohibits it. Who is allowed to move this amendment may be an open question, but I contend that it would be a Good Thing if it were moved.

Well, I really wasn't assuming much of anything, in particular I wasn't assuming that the process of amending the bylaws would involve the board at all, the way ordinary motions allegedly do. If I'm assuming anything, it's that there is a process for bylaws amendment, and that it must be followed.

I agree that it can be rescinded, but we disagree about what rescission means. In the Senate case the body apparently agreed to Rescind and Expunge from the Minutes. This is a strong statement that the motion of censure should never have been brought in the first place, which is what I contend rescission amounts to. Expunging expresses particularly vehement disapproval of the original action, and requires approval by a majority of the entire membership. This is not a case of a censure that is considered to have been appropriate at the time but has somehow "gone on long enough." (I continue to hold that no action is required in such cases. Once such a reprimand has been issued, the matter is done, and all parties move on.)
I believe at this point we both have a good idea of the outlines of the other's position. Unless you feel something can be achieved by further discussion, I'm done.

You'll have to check your bylaws, but this seems most unusual.
If the rules in RONR apply, annual meetings are meetings of the general membership. Board members, if present at all, are there as individuals, and the board as such is not present, as it is not in session. Unless your bylaws actually prohibit it, members are free to make motions at the annual meeting. If your bylaws do not expressly allow the practice, RONR does allow it. So there is no need to add permission to the bylaws since making motions is a basic right of membership. If the bylaws do prohibit the practice, you can simply amend them by striking that language from the bylaws.
Some organizations require prior review of member resolutions to be moved at the annual meeting, but this would have to be specified in detail in your bylaws, not simply because you have some other "practice" in place.

"Action" might include referring to a committee, postponing to the next meeting, or voting down a motion. Any actual decision that the assembly makes counts as "action". Motions don't simply die without any action at all. And if the meeting was adjourned while a motion was still pending, it will come up next time under Unfinished Business.
If no motion was ever made, no "item of business" actually occurred, so it's unlikely any decisions could have properly been made. While it's not very common, it's not unheard of for motions to be made during the Open Forum portion of a meeting, so it's possible that there would be action to record.
Additionally, the Open Forum can be used by members to give previous notice of intent to move something at the next meeting, either because notice is required, or because it lowers the vote threshold for adoption. Such notices are also recorded in the minutes.

Well, the chair under some conditions may adjourn a meeting without a motion, but being excessively angry is certainy not one of them. If the chair declares the meeting adjourned when members wish to continue meeting, then a member may immediately object (treating the prematurely announced adjournment as a unanimous consent request), or to raise a point of order that there was no motion to adjourn. Even if the chair announces adjournment because of a scheduled time to adjourn, the membership may move to set aside the orders of the day and continue meeting.
If a member immediately seeks recognition for any proper purpose, the chair is obligated to call the meeting back to order. If the chair refuses or simply departs, the vice-president/vice-chair, if present, presides over the remainder of the meeting, or no vice-officer is present, a chair pro-tem may be chosen.
But the OP seems to have abandoned the question, or lack thereof.

Ah, okay I misread that.
But the answer to the question, I think remains unchanged: The National rules have to be consulted to see if they apply to the local organization. If they don't apply, or don't cover the situation, then the rules in RONR apply.

There's no argument about that. No rights, including the right to vote, are remove by censure, and a motion remains in effect unless and until rescinded.
The quibble that remains is just what the rescision of a motion of censure implies. I contend that if not rescinded, this does not mean that the member remains under a cloud of censure forever. It means merely that the opinion that he did something wrong in the past was expressed.
We say "Sam was censured for a breach of decorum", not "Sam is under continuing censure for a breach of decorum in the late fifites."
It is Sam that is being censured because of some inappropriate action or failure to act. If Sam did nothing wrong, there is nothing to censure him for. We don't censure people just because we're feeling nasty. There has to be some event precipitating the censure.
If the assembly decides that the event was not worthy of censure, they can rescind the motion. If they believe it is "no longer" worthy of censure, they need not worry, since Sam, having endured a motion of censure, has already paid his debt to (the) society.

The board has the right to bring a recommended new logo to the membership at the annual meeting, and presumably they will find general members to move and second it. The general membership, once this is moved, has all the rights to adopt, reject, amend, postpone, refer, or otherwise dispose of the motion. The board cannot set rules for the annual meeting, and any proposal they bring to it, if the rules in RONR apply, is nothing more than a recommendation. If they recommend two choices instead of just one, there's no harm in asking the members to decide between them, as long as there is still a vote on whether or not to change the logo at all.
If a member believes that this "choose one" ballot is improper because it has not been approved by the membership, he can raise a Point of Order to that effect, and Appeal if necessary. Just remember that the board does not run the annual meeting. The board is not in session and so technically is not even present during the annual meeting, except possibly as individual members, with no more power than any other member.

Well, it is not in fact a board complaint unless the board voted to issue the complaint. A complaint from one board member is an individual complaint.
But it makes no difference in the answer, which is that there is no rule in RONR requiring the complainant to recuse himself.

Yes, it provides that any member could have moved to afford extra time to the speaker, but apparently everyone just sat there and said nothing. The book cannot jump up off the table and enforce itself.

You say that the winner will go against the current logo. So presumably after the decision has been made between the two, there will have to be another vote to substitute that new logo for the old. If that second vote fails, the current logo remains unchanged.

No, I don't think it does. If the assembly decided that Sam should be censured for a particular action, the disapproval of that action remains, even though Sam has been fully rehabilitated. The assembly might now believe Sam to be a great guy, even though it continues to believe that Sam's past action was improper. If they rescind the motion of censure, that would indicate that they have changed their collective mind and now believe that Sam acted properly back then, independent of whether they currently believe Sam to be a great guy or a bum.